Affine geometric method for linear programs

The algebraic (simplex) method is a complete enumerating algorithm to solve bounded linear programs (LP). It converts all the inequality constraints to equality to obtain a system of equations by introducing slack/surplus variables, converts all non-restricted (in sign) variables to restricted ones by substituting the difference of two new variables, and finally solves all of its square subsystems. This conversion of an LP into a pure algebraic version overlooks the original space of the decision variables and treats all variables alike throughout the process. We propose an effective three-phase algebraic method to overcome these deficiencies. Phase I concentrates on an early recognition of basic feasible solutions (BFS) which are located on the coordinate axis of each decision variable. Phase II finds the BFS which are on the faces of the coordinate system by solving some subsystems of equations which depend on the number of decision variables and constraints. Finally, in Phase III, it locates the BFS which are not on the faces of the coordinate system. The feasibility test is performed efficiently at the end of each phase. The algorithm works directly with the original variables even if some are unrestricted. The proposed method reduces considerably the computational complexity because it does not include any slack/surplus variables. The algorithmic strategy is to partition the set of BFS into three distinct sub-sets with common characteristics. Numerical examples illustrate the new solution algorithm.